The poll showed the Texas governor trailing Newt Gingrich and Ron Paul and well behind Herman Cain, who took the lead. Mitt Romney was a close second.

The poll comes as Mr. Perry is trying to re-energize his campaign, rolling out a tax plan that includes a flat-tax component, launching an ad blitz in Iowa and hiring a number of veteran GOP operatives to redirect his flagging campaign. Mr. Perry’s fortunes have tumbled in the weeks since he initially rocketed to front-runner status shortly after announcing his candidacy this summer.

In a prior CBS/New York Times survey, conducted last month, Mr. Perry led the GOP field with 23% support among likely primary voters.